The degrees of freedom of the Lasso for general design matrix

Details The degrees of freedom of the Lasso for general design matrix The degrees of freedom of the Lasso for general design matrix

2011-11-04

arxiv.org/abs/1111.1162v2

Mathematics

Statistics Theory

A short version appeared in SPARS'11, June 2011 Previously entitled "The degrees of freedom of penalized l1 minimization"

Scientific paper

In this paper, we study the degrees of freedom (df) of penalized l1 minimization (also known as the Lasso) for linear regression models. We give a closed-form expression of the degrees of freedom of the Lasso response. Namely, we show that for any given Lasso regularization parameter \lambda and any observed data y belongs to a set of full measure, the cardinal of the support of a particular solution of the Lasso problem is an unbiased estimator of the degrees of freedom of the Lasso response. This is achieved without any assumption on the uniqueness of the Lasso solution. Thus, our result remains true for both the underdetermined and the overdetermined case studied originally in Zou et al.. We also prove that a key result in Zou et al. is not true by providing a simple counterexample. An effective estimator of the number of degrees of freedom may have several applications including an objectively guided choice of the regularization parameter in the Lasso through the SURE framework.

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